The combined application of molecular markers and ecophysiological models can help to speed up the development of improved cultivars and enable the dissection of complex agronomic traits into the underlying physiological factors (e.g., for an efficient selection of candidate genes) (Boote et al., 2001; Hammer et al., 2006; Slafer, 2003). Connecting different levels of biological organization could provide significant impacts on crop improvement beyond single gene traits (Hammer et al., 2004). Dissecting complex traits into the parameters of ecophysiological models could be used for detecting quantitative trait loci (QTL) responsible for the phenotypic variability, while functional genome analysis provides information on the underlying genes and assists in the establishment of stable markers for marker-assisted selection. Theoretically, combining ecophysiological models with QTL models allows the prediction of the behavior of virtual plants of a breeding population under different environmental conditions, including conditions that do not exist or appear at present (Reymond et al., 2004). This could facilitate the in silico selection of favorable allele combinations (e.g., for climate change scenarios) by using computer simulations predicting the development of virtual genotypes under different environmental situations. QTL-based model approaches allow the prediction of any possible genotype of a segregating population in a wide range of environmental conditions (Tardieu, 2003). Because quantitative traits with genotype by environment interactions can be linked to stable underlying genotype characteristics, Yin et al. (2000) and Reymond et al. (2003) proposed that a genetic analysis could be carried out on parameters that describe the responses of genotypes to environmental conditions. In contrast to classical approaches that distinguish between constitutive QTL detected under any environmental condition, adaptive QTL describe adaptation processes and are detected only in specific (e.g., stress) environments. As a consequence adaptive QTL describe, in the present study, the adaptation process itself or the plasticity of a trait and are therefore directly detected on responses to environmental conditions. Genotypes independent of those used for QTL detection and for model parameterization have been used for testing the predictive quality of a combined QTL and ecophysiological model by Reymond et al. (2003).
Quantitative traits are inherently too complex to be described by a single value because their phenotypes change with age, metabolic rate, or environmental conditions (Ma et al., 2002). Due to their simplicity, linear models are widely used (e.g., to describe changes with age). However, even if the data show good linearity throughout large time intervals, they cannot be applied for data showing rate change points (Buchwald, 2007). Exponential, saturating, and sigmoidal functions have been established to describe the growth trajectories that include rate changes. Exponential and saturating functions describe extreme situations, which are generally not realistic during extended time periods. Sigmoidal functions combine both the situations and are well suited to describe most plant developmental processes. QTL analysis was successfully carried out on sigmoidal growth curves (Ma et al., 2002; Malosetti et al., 2006). However, the use of sigmoidal growth curves has several disadvantages: 1) large datasets representing a wide range of plant development stages are needed; 2) many of the widely used functions are somewhat inflexible due to fixed inflexion points, which is the case for the logistic function; and 3) more flexible ones like the Richards function (Richards, 1959) show high covariances between parameters, which leads to convergence problems and instable model parameters.
In the present study, we propose a two-phase linear model to describe preflowering leaf area (LA) development in B. oleracea. The advantages are two-fold. First, n-linear extensions can easily be generated. Parameterizable transitions between the linear segments by adding additional terms with successive rate-change points result in completely general models with n successive slopes, which are separated by n-1 rate-change points (Buchwald, 2007). Second, particular growth rates can be estimated for discrete durations, which may give information about the altered activity of QTL during different growth stages. To account for changing environmental conditions, the two-phase linear function describing LA development under optimum growth conditions was combined with a plateau function, which estimates the effects of drought on leaf expansion rates (LER). The plateau function is a special form of an n-phase linear function, including periods where the slope equals to zero. Two rate change points reflect the point when the stress effect starts to significantly change LER and the soil water status at which LER becomes zero.
Model parameters are given for each genotype by individual parameter estimates and based on particular QTL allele combinations in the present study. Many combined QTL and ecophysiological modeling approaches show pooled data from different levels of environmental stress (Reymond et al., 2003), sowing dates (Yin et al., 2005), or temperature regimes (Uptmoor et al., 2008). Earlier studies have shown that differences between genotypes in yield and biomass accumulation are only moderately predictive if all plants experienced the same environmental conditions. Yin et al. (2000) found in barley (Hordeum vulgare L.) RILs that correlations between predicted and measured values were mainly due to the segregation on one particular locus and that correlations were largely reduced when yield and biomass were predicted on independent datasets. One major reason for the low accuracy of crop models in predicting genotype effects may be that experimental errors blur trait differences between genotypes and that there is no substantial variability for all model input parameters in the segregating population. Therefore, we used a wide cross between chinese kale [B. oleracea var. alboglabra (L.H. Bailey) Musil] and broccoli (B. oleracea var. italica Plenck) segregating for both LER under well-watered conditions and adaptation to drought (Sebastian et al., 2000). Experiments for model parameterization and evaluation were carried out under controlled conditions to minimize random experimental errors. Prediction accuracies are shown separately for stress and nonstress conditions at different development stages. The main objectives of the present study were the parameterization of a model illustrating leaf area development of a B. oleracea doubled haploid (DH) population under well-watered and drought stress conditions individually for each DH line of the population, the detection of QTL on the parameters of the model followed by a model parameterization based on QTL effects, the evaluation of the model using independent experiments, and a comparison of the two methods used for model parameterization.
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